Differentiable manifold

A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the center and right charts, the Tropic of Cancer is a smooth curve, whereas in the left chart it has a sharp corner. The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable.

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.

In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their composition on chart intersections in the atlas must be differentiable functions on the corresponding linear space. In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the various charts to one another are called transition maps.

Differentiability means different things in different contexts including: continuously differentiable, k times differentiable, smooth, and holomorphic. Furthermore, the ability to induce such a differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields. Differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics, general relativity, and Yang–Mills theory. It is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry.

The emergence of differential geometry as a distinct discipline is generally credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture before the faculty at Göttingen.[1] He motivated the idea of a manifold by an intuitive process of varying a given object in a new direction, and presciently described the role of coordinate systems and charts in subsequent formal developments:

Having constructed the notion of a manifoldness of n dimensions, and found that its true character consists in the property that the determination of position in it may be reduced to n determinations of magnitude, ... – B. Riemann

A presentation of a topological manifold is a second countableHausdorffspace that is locally homeomorphic to a linear space, by a collection (called an atlas) of homeomorphisms called charts. The composition of one chart with the inverse of another chart is a function called a transition map, and defines a homeomorphism of an open subset of the linear space onto another open subset of the linear space. This formalizes the notion of "patching together pieces of a space to make a manifold" – the manifold produced also contains the data of how it has been patched together. However, different atlases (patchings) may produce "the same" manifold; a manifold does not come with a preferred atlas. Thus, one defines a topological manifold to be a space as above with an equivalence class of atlases, where one defines equivalence of atlases below.

There are a number of different types of differentiable manifolds, depending on the precise differentiability requirements on the transition functions. Some common examples include the following:

A differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition maps are all differentiable. More generally, a Ck-manifold is a topological manifold with an atlas whose transition maps are all k-times continuously differentiable.

A smooth manifold or C∞-manifold is a differentiable manifold for which all the transition maps are smooth. That is, derivatives of all orders exist; so it is a Ck-manifold for all k. An equivalence class of such atlases is said to be a smooth structure.

An analytic manifold, or Cω-manifold is a smooth manifold with the additional condition that each transition map is analytic: the Taylor expansion is absolutely convergent and equals the function on some open ball.

While there is a meaningful notion of a Ckatlas, there is no distinct notion of a Ckmanifold other than C0 (continuous maps: a topological manifold) and C∞ (smooth maps: a smooth manifold), because for every Ck-structure with k > 0, there is a unique Ck-equivalent C∞-structure (every Ck-structure is uniquely smoothable to a C∞-structure) – a result of Whitney.[6] In fact, every Ck-structure is uniquely smoothable to a Cω-structure. Furthermore, two Ck atlases that are equivalent to a single C∞ atlas are equivalent as Ck atlases, so two distinct Ck atlases do not collide. See Differential structure § Existence and uniqueness theorems for details. Thus, one uses the terms "differentiable manifold" and "smooth manifold" interchangeably; this is in stark contrast to Ckmaps, where there are meaningful differences for different k. For example, the Nash embedding theorem states that any manifold can be Ck isometrically embedded in Euclidean space RN – for any 1 ≤ k ≤ ∞ there is a sufficiently large N, but N depends on k.

On the other hand, complex manifolds are significantly more restrictive. As an example, Chow's theorem states that any projective complex manifold is in fact a projective variety – it has an algebraic structure.

Every topological manifold has an atlas. A Ck-atlas is an atlas whose transition maps are Ck. A topological manifold has a C0-atlas and in general a Ck-manifold has a Ck-atlas. A continuous atlas is a C0 atlas, a smooth atlas is a C∞ atlas and an analytic atlas is a Cω atlas. If the atlas is at least C1, it is also called a differential structure or differentiable structure. A holomorphic atlas is an atlas whose underlying Euclidean space is defined on the complex field and whose transition maps are biholomorphic.

Different atlases can give rise to, in essence, the same manifold. The circle can be mapped by two coordinate charts, but if the domains of these charts are changed slightly a different atlas for the same manifold is obtained. These different atlases can be combined into a bigger atlas. It can happen that the transition maps of such a combined atlas are not as smooth as those of the constituent atlases. If Ck atlases can be combined to form a Ck atlas, then they are called compatible. Compatibility of atlases is an equivalence relation; by combining all the atlases in an equivalence class, a maximal atlas can be constructed. Each Ck atlas belongs to a unique maximal Ck atlas.

The notion of a pseudogroup[7] provides a flexible generalization of atlases in order to allow a variety of different structures to be defined on manifolds in a uniform way. A pseudogroup consists of a topological space S and a collection Γ consisting of homeomorphisms from open subsets of S to other open subsets of S such that

If f ∈ Γ, and U is an open subset of the domain of f, then the restriction f|U is also in Γ.

If f is a homeomorphism from a union of open subsets of S, ∪iUi{\displaystyle \cup _{i}\,U_{i}}, to an open subset of S, then f ∈ Γ provided f|Ui∈Γ{\displaystyle f|_{U_{i}}\in \Gamma } for every i.

For every open U ⊂ S, the identity transformation of U is in Γ.

If f ∈ Γ, then f−1 ∈ Γ.

The composition of two elements of Γ is in Γ.

These last three conditions are analogous to the definition of a group. Note that Γ need not be a group, however, since the functions are not globally defined on S. For example, the collection of all local Ckdiffeomorphisms on Rn form a pseudogroup. All biholomorphisms between open sets in Cn form a pseudogroup. More examples include: orientation preserving maps of Rn, symplectomorphisms, Möbius transformations, affine transformations, and so on. Thus, a wide variety of function classes determine pseudogroups.

An atlas (Ui, φi) of homeomorphisms φi from Ui ⊂ M to open subsets of a topological space S is said to be compatible with a pseudogroup Γ provided that the transition functions φj ∘ φi−1 : φi(Ui ∩ Uj) → φj(Ui ∩ Uj) are all in Γ.

A differentiable manifold is then an atlas compatible with the pseudogroup of Ck functions on Rn. A complex manifold is an atlas compatible with the biholomorphic functions on open sets in Cn. And so forth. Thus, pseudogroups provide a single framework in which to describe many structures on manifolds of importance to differential geometry and topology.

Sometimes, it can be useful to use an alternative approach to endow a manifold with a Ck-structure. Here k = 1, 2, ..., ∞, or ω for real analytic manifolds. Instead of considering coordinate charts, it is possible to start with functions defined on the manifold itself. The structure sheaf of M, denoted Ck, is a sort of functor that defines, for each open set U ⊂ M, an algebra Ck(U) of continuous functions U → R. A structure sheaf Ck is said to give M the structure of a Ck manifold of dimension n provided that, for any p ∈ M, there exists a neighborhood U of p and n functions x1, ..., xn ∈ Ck(U) such that the map f = (x1, ..., xn) : U → Rn is a homeomorphism onto an open set in Rn, and such that Ck|U is the pullback of the sheaf of k-times continuously differentiable functions on Rn.[8]

In particular, this latter condition means that any function h in Ck(V), for V, can be written uniquely as h(x) = H(x1(x), ..., xn(x)), where H is a k-times differentiable function on f(V) (an open set in Rn). Thus, the sheaf-theoretic viewpoint is that the functions on a differentiable manifold can be expressed in local coordinates as differentiable functions on Rn, and a fortiori this is sufficient to characterize the differential structure on the manifold.

A similar, but more technical, approach to defining differentiable manifolds can be formulated using the notion of a ringed space. This approach is strongly influenced by the theory of schemes in algebraic geometry, but uses local rings of the germs of differentiable functions. It is especially popular in the context of complex manifolds.

We begin by describing the basic structure sheaf on Rn. If U is an open set in Rn, let

O(U) = Ck(U, R)

consist of all real-valued k-times continuously differentiable functions on U. As U varies, this determines a sheaf of rings on Rn. The stalk Op for p ∈ Rn consists of germs of functions near p, and is an algebra over R. In particular, this is a local ring whose unique maximal ideal consists of those functions that vanish at p. The pair (Rn, O) is an example of a locally ringed space: it is a topological space equipped with a sheaf whose stalks are each local rings.

A differentiable manifold (of class Ck) consists of a pair (M, OM) where M is a second countableHausdorffspace, and OM is a sheaf of local R-algebras defined on M, such that the locally ringed space (M, OM) is locally isomorphic to (Rn, O). In this way, differentiable manifolds can be thought of as schemes modelled on Rn. This means that [9] for each point p ∈ M, there is a neighborhood U of p, and a pair of functions (f, f#), where

f : U → f(U) ⊂ Rn is a homeomorphism onto an open set in Rn.

f#: O|f(U) → f∗ (OM|U) is an isomorphism of sheaves.

The localization of f# is an isomorphism of local rings

f#f(p) : Of(p) → OM,p.

There are a number of important motivations for studying differentiable manifolds within this abstract framework. First, there is no a priori reason that the model space needs to be Rn. For example, (in particular in algebraic geometry), one could take this to be the space of complex numbers Cn equipped with the sheaf of holomorphic functions (thus arriving at the spaces of complex analytic geometry), or the sheaf of polynomials (thus arriving at the spaces of interest in complex algebraic geometry). In broader terms, this concept can be adapted for any suitable notion of a scheme (see topos theory). Second, coordinates are no longer explicitly necessary to the construction. The analog of a coordinate system is the pair (f, f#), but these merely quantify the idea of local isomorphism rather than being central to the discussion (as in the case of charts and atlases). Third, the sheaf OM is not manifestly a sheaf of functions at all. Rather, it emerges as a sheaf of functions as a consequence of the construction (via the quotients of local rings by their maximal ideals). Hence, it is a more primitive definition of the structure (see synthetic differential geometry).

A final advantage of this approach is that it allows for natural direct descriptions of many of the fundamental objects of study to differential geometry and topology.

The cotangent space at a point is Ip/Ip2, where Ip is the maximal ideal of the stalk OM,p.

A real valued function f on an n-dimensional differentiable manifold M is called differentiable at a point p ∈ M if it is differentiable in any coordinate chart defined around p. In more precise terms, if {U,φ}{\displaystyle \{U,\varphi \}} is a chart where U{\displaystyle U} is an open set in M{\displaystyle M} containing p and φ:U→Rn{\displaystyle \varphi :U\to {\mathbf {R} }^{n}} is the map defining the chart, then f is differentiable if and only if

is differentiable at φ(p){\displaystyle \varphi (p)}, that is f is a differentiable function from the open set φ(U){\displaystyle \varphi (U)}, considered as a subset of Rn{\displaystyle {\mathbf {R} }^{n}}, to R{\displaystyle \mathbf {R} }. In general, there will be many available charts; however, the definition of differentiability does not depend on the choice of chart at p. It follows from the chain rule applied to the transition functions between one chart and another that if f is differentiable in any particular chart at p, then it is differentiable in all charts at p. Analogous considerations apply to defining Ck functions, smooth functions, and analytic functions.

There are various ways to define the derivative of a function on a differentiable manifold, the most fundamental of which is the directional derivative. The definition of the directional derivative is complicated by the fact that a manifold will lack a suitable affine structure with which to define vectors. Therefore, the directional derivative looks at curves in the manifold instead of vectors.

Given a real valued function f on an m dimensional differentiable manifold M, the directional derivative of f at a point p in M is defined as follows. Suppose that γ(t) is a curve in M with γ(0) = p, which is differentiable in the sense that its composition with any chart is a differentiable curve in Rm. Then the directional derivative of f at p along γ is

then, by the chain rule, f has the same directional derivative at p along γ1 as along γ2. This means that the directional derivative depends only on the tangent vector of the curve at p. Thus, the more abstract definition of directional differentiation adapted to the case of differentiable manifolds ultimately captures the intuitive features of directional differentiation in an affine space.

in every coordinate chart φ. Therefore, the equivalence classes are curves through p with a prescribed velocity vector at p. The collection of all tangent vectors at p forms a vector space: the tangent space to M at p, denoted TpM.

If X is a tangent vector at p and f a differentiable function defined near p, then differentiating f along any curve in the equivalence class defining X gives a well-defined directional derivative along X:

Once again, the chain rule establishes that this is independent of the freedom in selecting γ from the equivalence class, since any curve with the same first order contact will yield the same directional derivative.

If the function f is fixed, then the mapping

X↦Xf(p){\displaystyle X\mapsto Xf(p)}

is a linear functional on the tangent space. This linear functional is often denoted by df(p) and is called the differential of f at p:

One of the topological features of the sheaf of differentiable functions on a differentiable manifold is that it admits partitions of unity. This distinguishes the differential structure on a manifold from stronger structures (such as analytic and holomorphic structures) that in general fail to have partitions of unity.

Suppose that M is a manifold of class Ck, where 0 ≤ k ≤ ∞. Let {Uα} be an open covering of M. Then a partition of unity subordinate to the cover {Uα} is a collection of real-valued Ck functions φi on M satisfying the following conditions:

(Note that this last condition is actually a finite sum at each point because of the local finiteness of the supports of the φi.)

Every open covering of a Ck manifold M has a Ck partition of unity. This allows for certain constructions from the topology of Ck functions on Rn to be carried over to the category of differentiable manifolds. In particular, it is possible to discuss integration by choosing a partition of unity subordinate to a particular coordinate atlas, and carrying out the integration in each chart of Rn. Partitions of unity therefore allow for certain other kinds of function spaces to be considered: for instance Lp spaces, Sobolev spaces, and other kinds of spaces that require integration.

Suppose M and N are two differentiable manifolds with dimensions m and n, respectively, and f is a function from M to N. Since differentiable manifolds are topological spaces we know what it means for f to be continuous. But what does "f is Ck(M, N)" mean for k ≥ 1? We know what that means when f is a function between Euclidean spaces, so if we compose f with a chart of M and a chart of N such that we get a map that goes from Euclidean space to M to N to Euclidean space we know what it means for that map to be Ck(Rm, Rn). We define "f is Ck(M, N)" to mean that all such compositions of f with charts are Ck(Rm, Rn). Once again, the chain rule guarantees that the idea of differentiability does not depend on which charts of the atlases on M and N are selected. However, defining the derivative itself is more subtle. If M or N is itself already a Euclidean space, then we don't need a chart to map it to one.

For a Ck manifold M, the set of real-valued Ck functions on the manifold forms an algebra under pointwise addition and multiplication, called the algebra of scalar fields or simply the algebra of scalars. This algebra has the constant function 1 as the multiplicative identity, and is a differentiable analog of the ring of regular functions in algebraic geometry.

It is possible to reconstruct a manifold from its algebra of scalars, first as a set, but also as a topological space – this is an application of the Banach–Stone theorem, and is more formally known as the spectrum of a C*-algebra. First, there is a one-to-one correspondence between the points of M and the algebra homomorphisms φ: Ck(M) → R, as such a homomorphism φ corresponds a codimension one ideal in Ck(M) (namely the kernel of φ), which is necessarily a maximal ideal. On the converse, every maximal ideal in this algebra is an ideal of functions vanishing at a single point, which demonstrates that MSpec (the Max Spec) of Ck(M) recovers M as a point set, though in fact it recovers M as a topological space.

One can define various geometric structures algebraically in terms of the algebra of scalars, and these definitions often generalize to algebraic geometry (interpreting rings geometrically) and operator theory (interpreting Banach spaces geometrically). For example, the tangent bundle to M can be defined as the derivations of the algebra of smooth functions on M.

This "algebraization" of a manifold (replacing a geometric object with an algebra) leads to the notion of a C*-algebra – a commutative C*-algebra being precisely the ring of scalars of a manifold, by Banach–Stone, and allows one to consider noncommutative C*-algebras as non-commutative generalizations of manifolds. This is the basis of the field of noncommutative geometry.

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The tangent space of a point consists of the possible directional derivatives at that point, and has the same dimensionn as does the manifold. For a set of (non-singular) coordinates xk local to the point, the coordinate derivatives ∂k=∂∂xk{\displaystyle \partial _{k}={\frac {\partial }{\partial x_{k}}}} define a holonomic basis of the tangent space. The collection of tangent spaces at all points can in turn be made into a manifold, the tangent bundle, whose dimension is 2n. The tangent bundle is where tangent vectors lie, and is itself a differentiable manifold. The Lagrangian is a function on the tangent bundle. One can also define the tangent bundle as the bundle of 1-jets from R (the real line) to M.

One may construct an atlas for the tangent bundle consisting of charts based on Uα × Rn, where Uα denotes one of the charts in the atlas for M. Each of these new charts is the tangent bundle for the charts Uα. The transition maps on this atlas are defined from the transition maps on the original manifold, and retain the original differentiability class.

The dual space of a vector space is the set of real valued linear functions on the vector space. The cotangent space at a point is the dual of the tangent space at that point, and the cotangent bundle is the collection of all cotangent spaces.

Like the tangent bundle, the cotangent bundle is again a differentiable manifold. The Hamiltonian is a scalar on the cotangent bundle. The total space of a cotangent bundle has the structure of a symplectic manifold. Cotangent vectors are sometimes called covectors. One can also define the cotangent bundle as the bundle of 1-jets of functions from M to R.

Elements of the cotangent space can be thought of as infinitesimal displacements: if f is a differentiable function we can define at each point p a cotangent vector dfp, which sends a tangent vector Xp to the derivative of f associated with Xp. However, not every covector field can be expressed this way. Those that can are referred to as exact differentials. For a given set of local coordinates xk, the differentials dxkp form a basis of the cotangent space at p.

The tensor bundle is not a differentiable manifold in the traditional sense, since it is infinite dimensional. It is however an algebra over the ring of scalar functions. Each tensor is characterized by its ranks, which indicate how many tangent and cotangent factors it has. Sometimes these ranks are referred to as covariant and contravariant ranks, signifying tangent and cotangent ranks, respectively.

A frame (or, in more precise terms, a tangent frame), is an ordered basis of particular tangent space. Likewise, a tangent frame is a linear isomorphism of Rn to this tangent space. A moving tangent frame is an ordered list of vector fields that give a basis at every point of their domain. One may also regard a moving frame as a section of the frame bundle F(M), a GL(n, R)principal bundle made up of the set of all frames over M. The frame bundle is useful because tensor fields on M can be regarded as equivariant vector-valued functions on F(M).

On a manifold that is sufficiently smooth, various kinds of jet bundles can also be considered. The (first-order) tangent bundle of a manifold is the collection of curves in the manifold modulo the equivalence relation of first-order contact. By analogy, the k-th order tangent bundle is the collection of curves modulo the relation of k-th order contact. Likewise, the cotangent bundle is the bundle of 1-jets of functions on the manifold: the k-jet bundle is the bundle of their k-jets. These and other examples of the general idea of jet bundles play a significant role in the study of differential operators on manifolds.

The notion of a frame also generalizes to the case of higher-order jets. Define a k-th order frame to be the k-jet of a diffeomorphism from Rn to M.[10] The collection of all k-th order frames, Fk(M), is a principal Gk bundle over M, where Gk is the group of k-jets; i.e., the group made up of k-jets of diffeomorphisms of Rn that fix the origin. Note that GL(n, R) is naturally isomorphic to G1, and a subgroup of every Gk, k ≥ 2. In particular, a section of F2(M) gives the frame components of a connection on M. Thus, the quotient bundle F2(M) / GL(n, R) is the bundle of symmetric linear connections over M.

Many of the techniques from multivariate calculus also apply, mutatis mutandis, to differentiable manifolds. One can define the directional derivative of a differentiable function along a tangent vector to the manifold, for instance, and this leads to a means of generalizing the total derivative of a function: the differential. From the perspective of calculus, the derivative of a function on a manifold behaves in much the same way as the ordinary derivative of a function defined on a Euclidean space, at least locally. For example, there are versions of the implicit and inverse function theorems for such functions.

There are, however, important differences in the calculus of vector fields (and tensor fields in general). In brief, the directional derivative of a vector field is not well-defined, or at least not defined in a straightforward manner. Several generalizations of the derivative of a vector field (or tensor field) do exist, and capture certain formal features of differentiation in Euclidean spaces. The chief among these are:

The Lie derivative, which is uniquely defined by the differential structure, but fails to satisfy some of the usual features of directional differentiation.

An affine connection, which is not uniquely defined, but generalizes in a more complete manner the features of ordinary directional differentiation. Because an affine connection is not unique, it is an additional piece of data that must be specified on the manifold.

Differentiable functions between two manifolds are needed in order to formulate suitable notions of submanifolds, and other related concepts. If f : M → N is a differentiable function from a differentiable manifold M of dimension m to another differentiable manifold N of dimension n, then the differential of f is a mapping df : TM → TN. It is also denoted by Tf and called the tangent map. At each point of M, this is a linear transformation from one tangent space to another:

Usually the rank of a function is a pointwise property. However, if the function has maximal rank, then the rank will remain constant in a neighborhood of a point. A differentiable function "usually" has maximal rank, in a precise sense given by Sard's theorem. Functions of maximal rank at a point are called immersions and submersions:

If m ≤ n, and f : M → N has rank m at p ∈ M, then f is called an immersion at p. If f is an immersion at all points of M and is a homeomorphism onto its image, then f is an embedding. Embeddings formalize the notion of M being a submanifold of N. In general, an embedding is an immersion without self-intersections and other sorts of non-local topological irregularities.

If m ≥ n, and f : M → N has rank n at p ∈ M, then f is called a submersion at p. The implicit function theorem states that if f is a submersion at p, then M is locally a product of N and Rm−n near p. In formal terms, there exist coordinates (y1, ..., yn) in a neighborhood of f(p) in N, and m − n functions x1, ..., xm−n defined in a neighborhood of p in M such that

The bundle of differential forms, at each point, consists of all totally antisymmetricmultilinear maps on the tangent space at that point. It is naturally divided into n-forms for each n at most equal to the dimension of the manifold; an n-form is an n-variable form, also called a form of degree n. The 1-forms are the cotangent vectors, while the 0-forms are just scalar functions. In general, an n-form is a tensor with cotangent rank n and tangent rank 0. But not every such tensor is a form, as a form must be antisymmetric.

This map is the one that relates covectors to infinitesimal displacements, mentioned above; some covectors are the exterior derivatives of scalar functions. It can be generalized into a map from the n-forms onto the (n+1)-forms. Applying this derivative twice will produce a zero form. Forms with zero derivative are called closed forms, while forms that are themselves exterior derivatives are known as exact forms.

The space of differential forms at a point is the archetypal example of an exterior algebra; thus it possesses a wedge product, mapping a k-form and l-form to a (k + l)-form. The exterior derivative extends to this algebra, and satisfies a version of the product rule:

From the differential forms and the exterior derivative, one can define the de Rham cohomology of the manifold. The rank n cohomology group is the quotient group of the closed forms by the exact forms.

Every topological manifold in dimension 1, 2, or 3 has a unique differential structure (up to diffeomorphism); thus the concepts of topological and differentiable manifold are distinct only in higher dimensions. It is known that in each higher dimension, there are some topological manifolds with no smooth structure, and some with multiple non-diffeomorphic structures.

Every second-countable 1-manifold without boundary is homeomorphic to a disjoint union of countably many copies of R (the real line) and S (the circle); the only connected examples are R and S, and of these only S is compact. In higher dimensions, classification theory normally focuses only on compact connected manifolds.

For a classification of 2-manifolds, see surface: in particular compact connected oriented 2-manifolds are classified by their genus, which is a nonnegative integer.

The classification of n-manifolds for n greater than three is known to be impossible, even up to homotopy equivalence. Given any finitely presented group, one can construct a closed 4-manifold having that group as fundamental group. Since there is no algorithm to decide the isomorphism problem for finitely presented groups, there is no algorithm to decide whether two 4-manifolds have the same fundamental group. Since the previously described construction results in a class of 4-manifolds that are homeomorphic if and only if their groups are isomorphic, the homeomorphism problem for 4-manifolds is undecidable. In addition, since even recognizing the trivial group is undecidable, it is not even possible in general to decide whether a manifold has trivial fundamental group, i.e. is simply connected.

However, the situation becomes more tractable for simply connected smooth manifolds of dimension ≥ 5, where the h-cobordism theorem can be used to reduce the classification to a classification up to homotopy equivalence, and surgery theory can be applied.[14] This has been carried out to provide an explicit classification of simply connected 5-manifolds by Dennis Barden.

A Riemannian manifold is a differentiable manifold on which the tangent spaces are equipped with inner products in a differentiable fashion. The inner product structure is given in the form of a symmetric 2-tensor called the Riemannian metric. This metric can be used to interconvert vectors and covectors, and to define a rank 4 Riemann curvature tensor. On a Riemannian manifold one has notions of length, volume, and angle. Any differentiable manifold can be given a Riemannian structure.

The category of smooth manifolds with smooth maps lacks certain desirable properties, and people have tried to generalize smooth manifolds in order to rectify this. Diffeological spaces use a different notion of chart known as a "plot". Frölicher spaces and orbifolds are other attempts.

A rectifiable set generalizes the idea of a piece-wise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in general manifolds.

^This form is clearly nondegenerate, and it must be closed because it is top-dimensional with respect to the surface; this reflects the exceptional isomorphism of Lie groups Sp(2, R) ≅ SL(2, R) between the symplectic group (corresponding to symplectic structure) and the special linear group (corresponding to orientable structure). Note that a symplectic structure requires an additional integrability condition, beyond this isomorphism of groups: it is not just a G-structure.